Theorem elcncf1di 12724

Description: Membership in the set of continuous complex functions from A to B. (Contributed by Paul Chapman, 26-Nov-2007.)

Hypotheses

Ref	Expression
elcncf1d.1	|- ( ph -> F : A --> B )
elcncf1d.2	|- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
elcncf1d.3	|- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )

Assertion

Ref	Expression
elcncf1di	|- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )

Distinct variable groups:   x, w, y, A   w, B, x, y   w, F, x, y   ph, w, x, y   w, Z

Allowed substitution hints:   Z( x, y)

Proof of Theorem elcncf1di

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elcncf1d.1	. . 3 |- ( ph -> F : A --> B )
2		elcncf1d.2	. . . . . 6 |- ( ph -> ( ( x e. A /\ y e. RR+ ) -> Z e. RR+ ) )
3	2	imp 123	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> Z e. RR+ )
4		an32 551	. . . . . . . . 9 |- ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) <-> ( ( x e. A /\ y e. RR+ ) /\ w e. A ) )
5	4	anbi2i 452	. . . . . . . 8 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) <-> ( ph /\ ( ( x e. A /\ y e. RR+ ) /\ w e. A ) ) )
6		anass 398	. . . . . . . 8 |- ( ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) <-> ( ph /\ ( ( x e. A /\ y e. RR+ ) /\ w e. A ) ) )
7	5, 6	bitr4i 186	. . . . . . 7 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) <-> ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) )
8		elcncf1d.3	. . . . . . . 8 |- ( ph -> ( ( ( x e. A /\ w e. A ) /\ y e. RR+ ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
9	8	imp 123	. . . . . . 7 |- ( ( ph /\ ( ( x e. A /\ w e. A ) /\ y e. RR+ ) ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
10	7, 9	sylbir 134	. . . . . 6 |- ( ( ( ph /\ ( x e. A /\ y e. RR+ ) ) /\ w e. A ) -> ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
11	10	ralrimiva 2503	. . . . 5 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> A. w e. A ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
12		breq2 3928	. . . . . 6 |- ( z = Z -> ( ( abs ` ( x - w ) ) < z <-> ( abs ` ( x - w ) ) < Z ) )
13	12	rspceaimv 2792	. . . . 5 |- ( ( Z e. RR+ /\ A. w e. A ( ( abs ` ( x - w ) ) < Z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
14	3, 11, 13	syl2anc 408	. . . 4 |- ( ( ph /\ ( x e. A /\ y e. RR+ ) ) -> E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
15	14	ralrimivva 2512	. . 3 |- ( ph -> A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) )
16	1, 15	jca 304	. 2 |- ( ph -> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) )
17		elcncf 12718	. 2 |- ( ( A C_ CC /\ B C_ CC ) -> ( F e. ( A -cn-> B ) <-> ( F : A --> B /\ A. x e. A A. y e. RR+ E. z e. RR+ A. w e. A ( ( abs ` ( x - w ) ) < z -> ( abs ` ( ( F ` x ) - ( F ` w ) ) ) < y ) ) ) )
18	16, 17	syl5ibrcom 156	1 |- ( ph -> ( ( A C_ CC /\ B C_ CC ) -> F e. ( A -cn-> B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   A.wral 2414   E.wrex 2415   C_ wss 3066   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   CCcc 7611   < clt 7793   - cmin 7926   RR+crp 9434   abscabs 10762   -cn->ccncf 12715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-map 6537  df-cncf 12716

This theorem is referenced by:  elcncf1ii  12725  cncfmptc  12740  cncfmptid  12741  addccncf  12744  negcncf  12746